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Modularity of Social Networks
Image Ramki (www.wildventures.com)
Sitabhra SinhaThe Institute of Mathematical
Sciences, Chennai
Image R K Pan
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Outline or Why ? What ? How ?

• Why networks ?
• What structures underlie real life networks ?
• How do such structures affect network dynamics ?
• In particular,
• How does it affect ordering dynamics in spin
  models ?
• Significant for understanding social coordination
  processes such as consensus formation adoption
  of innovations

R K Pan SS, EPL (2009)
S Dasgupta, R K Pan SS, PRE Rapid (2009)
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Nodes
Why Networks ?
Links

• In a network of interacting components
• Emergence
• of qualitatively different behavior from
  individual components.
• E.g., component neuron, system brain
• Interactions add a new layer of complexity!
• The aim of studying networks
• how interactions ? complexity at systems-level

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Ubiquity of Networks
Networks appear at all scales
Proteins
Social contact
Food webs
Neuronal communication
Intra-cellular signalling
10-3 m
10-6 m
10-9 m
1 m
103 m
106 m
Molecules
Cells
Organisms
Populations
Ecologies
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Theoretical understanding of networks

• Regular lattice or grid (Physics)
• average path length N (no. of nodes)
• clustering high
• delta function distribution of degree
  (links/node)
• Random networks (Graph theory)
• average path length log N
• clustering low
• Poisson distribution of degree

Empirical networks are not random many have
certain structural patterns
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Example small-world networks
Low clustering, Short path length
High clustering, Large path length
High clustering, Short path length
Regular Network
Random Network
Small-world Network
p 1
0 lt p lt 1
p 0
Increasing Randomness
p fraction of random, long-range connections
Watts and Strogatz (1998) Many biological,
technological and social networks have connection
topologies that lie between the two extremes of
completely regular and completely random.
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Small-world networks can be highly clustered
(like regular networks ), yet have small
characteristic path lengths (as in random
networks ).
Characteristic path length ( l ( p ) ) and
clustering coefficient (C( p ) ) as network
randomness increases.
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A prominent example of small-world networks
Social networks

• Nodes individuals (N 3 109)
• Links social interactions (lt 150/node)
• Represent the underlying social
• network (e.g. a knows b)
• Approximated by real interactions
• between a b (e.g. through phone calls,
  emails, trades, )
• Weight represents interaction
• strength

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Example of social interaction network
Collaboration Networks
Authors
IMSc Physics co-authorship network (2000-08)
1
2
3
4
5
D
C
B
A
Papers
Bipartite Paper-Author network
Co-authorship network
1
2
4
5
3
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Re-constructing social networks Example Macaque
troupe social network
UAS-GKVK Campus, Bangalore Data Anindya Sinha
(NIAS,Blore)
The Bonnet Macaque (Macaca radiata) seen widely
in southern India
Usually live in large ( 40) multi-male,
multi-female troops where the adult individuals
( 10) develop strong affiliative relationships
Image Ramki (www.wildventures.com)
Image Arunkumar
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Macaque Social Networks can be defined in terms
ofgrooming frequencytotal grooming time
approach frequency
Data 1993-1997
grooming frequency
Numbers refer to rank among the adult females
from 11 (most dominant) to 1 (least dominant)
grooming time
approach frequency
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Network analysis can predict group dynamics !

• Female bonnet macaques
• usually remain in the group throughout their life
• as adults, form strong linear matrilineal
  dominance hierarchies that are stable over time

• Male bonnet macaques
• as adults, form unstable dominance hierarchies
• occupy low ranks when young, high when mature and
  at peak of health

Community detection generates consistent
partitions for females, not for males
Predictive power Observation in 1998 showed the
group had split into two (11,10,9,8,7,2) and
(6,5,4,3) 1 had died
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Importance of SW social network structure
Contagion propagation in society through contact
Spread of SARS from Taiwan, 2003
The small-worlds of public health (CDC
director)
Chen et al, Lect Notes Comp Sci 4506 (2009) 23
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Worldwide spread of SARS through international
airline network
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Why small-world pattern in complex networks at
all ?
R K Pan and SS, EPL 2009
We have shown that structurally
Watts-Strogatz network

Modular network
All the classic small-world structural
properties of Watts-Strogatz small world
networks e.g., high clustering, short average
distance, etc. are also seen in Modular networks
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Modular Networks dense connections within
certain sub-networks (modules) relatively few
connections between modules
Modules A mesoscopic organizational principle of
networks
Going beyond motifs but more detailed than global
description (L, C etc.)
Micro
Meso
Macro
Kim Park, WIREs Syst Biol Med, 2010
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Ubiquity of modular networks
Modular Biology (Hartwell et al, Nature
1999) Functional modules as a critical level of
biological organization
Modules in biological networks are often
associated with specific functions
Metabolic network of E coli (Guimera Amaral)
Chesapeake Bay foodweb (Ulanowicz et al)
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How about social small-world networks ?
Lets look at real re-constructed social contact
patterns A mobile phone interaction network
Onnela et al. PNAS,104,7332 (2007)

• Data
• A mobile phone operator in an European country,
  20 coverage
• Aggregated from a period of 18 weeks
• gt 7 million private mobile phone subscriptions
• Voice calls within the operator
• Require reciprocity of calls for a link
• Quantify tie strength (link weight)

Aggregate call duration Total number of calls
7 min
15 min (3 calls)
5 min
3 min
J-P Onnela
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Reconstructed network
Onnela et al. PNAS,104,7332 (2007)
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Modularity of social networks
4.6 ?106 nodes 7.0 ? 106 links

• Modularity Cohesive groups
• communities with dense internal sparse external
  connections
• Other examples of modular social networks
• Scientific collaborators
• e-mail communication
• PGP encryption web-of-trust
• non-human animals

Onnela et al. PNAS,104,7332 (2007)
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Q. What structure ? Ans. Modular
NextHow do such structures affect dynamics ?

• Over social networks, such dynamics can be of
• information or epidemic spreading
• consensus or opinion formation
• adoption of innovations

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A simple model of modular networks
Module random network
Adjacency matrix
Model parameter r Ratio of inter- to
intra-modular connectivity
The modularity of the network is changed keeping
avg degree constant
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Comparison with Watts-Strogatz model
Structural measures used
Communication efficiency
E avg path length, l -1 2 /N(N-1) ?igtjdij
Clustering coefficient
C fraction of observed to potential triads
(1 /N) ?i2ni / ki (ki - 1)
WS and Modular networks behave similarly as
function of p or r (Also for between-ness
centrality, edge clustering, etc)
In fact, for same N and ltkgt, we can find p and r
such that the WS and Modular networks have the
same modularity Q
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Then how can you tell them apart ?
Dynamics on Watts-Strogatz network different from
that on Modular networks
Network topology
Consider ordering or alignment of orientation on
such networks e.g., Ising spin model dynamics
minimizes H - ? Jij Si (t) Sj (t)
2 distinct time scales in Modular networks t
modular t global
Global order
Time required for global ordering diverges as r ?0
Modular order
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Universality
Almost identical two time scale behavior is seen
for oscillator synchronization (viz., nonlinear
relaxation oscillators)
Network topology
Consider synchronization on modular
networks e.g., Kuramoto oscillators d?i /dt w
(1/ki)?Kij sin (?j - ? i)
2 distinct time scales in Modular networks t
modular t global
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Eigenvalue spectra of the Laplacian
Shows the existence of spectral gap ? distinct
time scales
Modular network Laplacian spectra
Spectral gap in modular networks diverges with
decreasing r
gap
No gap
WS network Laplacian spectra
Existence of distinct time-scales in Modular
networks
No such distinction in Watts-Strogatz small-world
networks
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Diffusion process on modular networks
Random walker moving from one node to randomly
chosen nghbring node

• Also shows the existence of 2 distinct time
  scales
• fast intra-modular diffusion
• slower inter-modular diffusion

Relevant for diffusion of innovation or epidemics
Localization of eigenmodes of transition matrix
Distrn of first passage times for rnd walks to
reach a target node starting from a source node
ER
WS
Modular
Between modules diffusion
Within modules diffusion
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How about real small-world networks ?
The networks of cortical connections in mammalian
brain have been shown to have small-world
structural properties Our analysis reveals their
dynamical properties to be consistent with
modular small-world networks
gap
gap
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We again turn attention back to consensus
formation dynamics
Q. How does individual behavior at micro-level
relate to social phenomena at macro level ?
Order-disorder transitions in Social Coordination
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The emergence of novel phase of collective
behavior
Spin models of statistical physics simple models
of coordination or consensus formation

• Spin orientation mutually exclusive choices

• Choice dynamics decision based on information
  about choice of majority in local neighborhood

Simplest case 2 possible choices
Ising model with FM interactions each agent can
only be in one of 2 states (Yes/No or /-)
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Types of possible order in modular network of
Ising spins
Modular order
Global order
N spins, nm modules
FM interactions J gt 0
Avg magnetic moment / module
Total or global magnetic moment
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But how do the different ordered phases occur as
a function of modularity parameter r and
temperature T ? Long transient to global order
can be mistaken as modular order
Possible Phase Diagrams
Tgc
Tgc
Tgc
T
T
T
r
0
1
r
0
1
r
0
1
Can modular order be seen as a phase at all ?
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fraction of up spins in module
Magnetic moment of a single module
fraction of modules with
At eqlbm, for strong modularity (r ltlt 1)
Total magnetic moment
Minimizing free energy w.r.t. f
Continuous transition to modular order phase
below
Modular critical temp
links within module
As T is lowered, Another continuous transition
to global order phase below
Global critical temp
conn prob betn modules
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Phase diagram two transitions
r 0.002
T
There will be a phase corresponding to modular
but no global order (coexistence of contrary
opinions) even when all mutual interactions are
FM (favor consensus) !
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Even when global order is possible
Divergence of relaxation time with modularity
To switch from to - , module crosses
free energy barrier
Relaxation time to global order
-

At low T
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Thus, even when strongly modular network takes
very long to show global order
?Time required to achieve consensus increases
rapidly for a strongly modular social organization
But then How do certain innovations get adopted
rapidly ?

• Possible modifications to the dynamics
• Positive feedback
• Different strengths for inter/intra couplings

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I. The effect of Positive Feedback
Brian Arthur
The case of counter-clockwise clocks
Microsoft the 7 Dwarves
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The effect of Positive Feedback
Introduce a field H hM (proportional to
magnetization)

• Effectively increases inter-modular interactions
• Drives system away from critical line by
  increasing Tgc
• reduces

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II. Varying Strength of Coupling Between Within
Modules
Marc Granovetter
J0 strength of inter-modular connections
Ji strength of intra-modular connections
increasing J0 / Ji ? increasing r
Non-monotonic behavior of relaxation time vs
ratio of strengths of short- long-range
couplings Not seen in Watts-Strogatz SW
networks (Jeong et al, PRE 2005)
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Ongoing and Future work
Effect of modular contact structure in formal
games - Is it easier for cooperation to emerge
for Prisoners Dilemma on modular networks ?
Could modularity in social networks have arisen
as a result of modules promoting cooperation
necessary for building social organization ?
Role of multiple (gt2) choices q-state Potts
model
Having different types of interactions spin
glass behavior on modular networks
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Question Why Modular Networks ?
Suggestion modularity imparts robustness
E.g.,Variano et al, PRL 2004
Not quite ! Consider stability of a random
network with a modular structure.
N256
As random networks are divided into more modules
(m) they become more unstable
however, we see modular networks all around us.
Why ?
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Question Why Modular Networks ?
Clue Many of these modular networks also
possess multiple hubs !
Most real networks have non-trivial degree
distribution Degree total number of
connections for a node
Hubs nodes having high degree relative to other
nodes
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Why Modular Networks ? R K Pan SS, PRE 76
045103(R) (2007)

• Hypothesis Real networks optimize between
  several constraints,
• Minimizing link cost, i.e., total links L
• Minimizing average path length l
• Minimizing instability ?max

Minimizing link cost and avg path length yields
a star-shaped network with single hub
But unstable ! Instability measured by ?max
?max degree (i.e., degree of the hub) In fact,
for star network, ?max ?N
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How to satisfy all three constraints ?
Answer go modular !
As star-shaped networks are divided into more
modules they become more stable as stability
increases by decreasing the degree of hub nodes
?max ?N/m
Increasing stability, average path length
? 0.78
? 0.4
? 1
N 64, L N-1
Shown explicitly by Network Optimization Fix
link cost to min (LN-1) and minimize the energy
fn E (?) ? l (1- ?) ?max ? ?0,1
relative importance of path length constraint
over stability constraint
Transition to star configuration
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The robustness of modular structures
We have considered dynamical instability
criterion for network robustness how about
stability against structural perturbations
? E.g., w.r.t. random or targeted removal of
nodes
Scale-free network robust w.r.t. random
removal Random network robust w.r.t. targeted
removal Is the modular network still optimal ?
Surprisingly YES !
At the limit of extremely small L, optimal
modular networks networks with bimodal degree
distribution has been shown to be robust
w.r.t. targeted as well as random removal of nodes
Tanizawa et al, PRE 2005
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Similar mechanisms explain the emergence of
hierarchical structures in society
Many complex networks that we see in society (and
also nature) have hierarchical structures. Why ?
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Emergence of hierarchy

• Consider social networks as solutions to
  multi-constraint problems
• Increase communication efficiency ? Decrease
  average path length l in a network
• Minimize information load at each node ?
  Decrease max degree kmax
• Minimize overall communication needs ? Decrease
  total number of links L

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Emergence of hierarchy

• Minimize the energy function,
• E a l g L (1-a-g) kmax
• For a 0, g 1 optimal network is a chain
• For a 1, g 0, optimal network is a clique
• For a g 0.5, optimal network is a star
• Over large range of values of a and g,
  hierarchical networks emerge hierarchy measured
  by HQ

Cycles
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Hierarchical network
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Conclusions

• Modular networks are small-world networks
  indistinguishable from WS model generated
  networks by using structural measures
• Dynamics on modular networks (but not in WS
  networks) show distinct, separate time-scales
  manifested as Laplacian spectral gap seen in real
  networks (cortico-cortical brain networks)
• Collective behavior such as ordering in spin
  models show a novel phase corresponding to
  modular order, in addition to global order and no
  order even when mutual interactions favor
  consensus

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Thanks
Raj Kumar Pan
Sumithra Surendralal
Abhishek Dasgupta
Subinay Dasgupta
Anindya Sinha